Genuinely nonlocal product bases: Classification and entanglement-assisted discrimination

An orthogonal product basis of a composite Hilbert space is genuinely nonlocal if the basis states are locally indistinguishable across every bipartition. From an operational point of view, such a basis corresponds to a separable measurement that cannot be implemented by local operations and classical communication unless all the parties come together in a single location. In this work we classify genuinely nonlocal product bases into different categories. Our classification is based on the state elimination property of the set via orthogonality-preserving measurements when all the parties are spatially separated or different subsets of the parties come together. We then study local state discrimination protocols for several such bases with additional entangled resources shared among the parties. Apart from consuming less entanglement than teleportation-based schemes, our protocols indicate operational significance of the proposed classification and exhibit nontrivial use of genuine entanglement in the local state discrimination problem.